Help with parametric equations

In summary, the parametric equations for the semicircle x^2+y^2=a^2, y>0 using the slope t=dy/dx of the tangent at (x,y) as the parameter are x=asint and y=acost for 0<=t<=pi. To reduce the slope expression, we can use the relation a^2-x^2=b^2, where b is the radius of the circle. Then, we can express x as a function of s and y, and use this to eliminate x in another equation to get a parameterization for y. We can repeat this process with y as a function of s and x to get a parameterization for x.
  • #1
miglo
98
0

Homework Statement


find parametric equations for the semicircle x^2+y^2=a^2, y>0
using as parameter the slope t=dy/dx of the tangent to the curve at (x,y)


Homework Equations





The Attempt at a Solution


well i know that to parametrize a circle i would use x=acost and y=asint for 0<=t<=2pi
so i guess if i just want the top half i would use 0<=t<=pi ? but I am not sure how to use the derivative of y=sqrt(a^2-x^2) to parametrize the curve
i went ahead and found dy/dx to be 1/2(a^2-x^2)^(-1/2)*-2x or -x/sqrt(a^2-x^2)
i don't know what to do next...
 
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  • #2
Your expression for the slope is correct, but it will probably help you to reduce it a bit more (hint what is a2-x2?).

If we call the slope for s (our parameter), can you then express x as a function of s and y? If so, perhaps you can use this relation to get rid of x in another equation so it ends up only relating y and s, thus allowing you to find y as a function of s. Repeat with y as a function of s and x to get a parameterization for x.
 

What are parametric equations?

Parametric equations are a way to represent a set of equations using parameters. These parameters are usually represented by variables, and they describe the behavior of a function in terms of time or another independent variable.

Why are parametric equations useful?

Parametric equations are useful because they allow us to describe complex geometric shapes and curves using simple equations. They also make it easier to analyze and manipulate these shapes and curves, and they are especially helpful in computer graphics and animation.

How do you graph parametric equations?

To graph parametric equations, you can plot points on a coordinate plane by plugging in different values for the parameter. These points will form a curve, and by connecting them, you can see the shape of the graph. You can also use a graphing calculator or computer software to graph parametric equations.

What is the difference between parametric equations and Cartesian equations?

The main difference between parametric equations and Cartesian equations is that parametric equations describe a curve or shape in terms of parameters, while Cartesian equations describe it in terms of x and y coordinates. Parametric equations are also more versatile and can describe curves that Cartesian equations cannot.

How are parametric equations used in real life?

Parametric equations are used in many real-life applications, such as designing roller coasters, creating animations in movies and video games, and modeling the motion of objects in physics and engineering. They are also used in fields like economics, biology, and finance to analyze and predict behavior and trends.

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